Wikipedia:Reference desk/Archives/Mathematics/2006 October 6

Arithmetic problem related to internet
The question has been moved to Computing/IT reference desk

Graph and fractals
Are graphs (in the sense of graph theory, not plots) also fractals???


 * No, they are incomparable entities. Graphs are abstract structures; if you see a picture of a graph it is because someone decided to represent the vertices of the graph as blobs and the edges as curve segments. A graph can also be represented as a connectivity matrix or in several other ways. Inasmuch as a graph depiction has a shape, it is typically the result of rather arbitrary choices. A fractal, on the other hand, is a shape in some space – usually the plane, but you can also have fractals in 3D space and other dimensions. Normally the shape used for depicting a graph will not be a fractal, but for some infinite graphs that may be possible. --Lambiam Talk  17:44, 6 October 2006 (UTC)


 * You might use a graph representation in a computer progrthe sameam to create the fractal though... - Rainwarrior 19:11, 6 October 2006 (UTC)


 * A hallmark of fractals is that they look similar no matter at what magnification you look at them. This is known as scale invariance, and this phenomenon can indeed also be observed for graphs (or more precisely: for families of graphs growing to unbounded sizes). See scale-free network. Also, look at L-system for some nice pictures of fractals that at least look like graph embeddings. Simon A. 19:37, 6 October 2006 (UTC)

I remember going to see a talk on A New Kind of Science. I remember Wolfram wants to define graphs as the fundamental things in nature, and the dimensionality of space by some kind of scaling law:
 * $$n^{-1}\log(\#\mbox{ of vertices seperated by at most }n\mbox{ edges from some fiducial vertex }f)\to d\qquad\mbox{as}\qquad n\to\infty$$

Presumably graph theorists have a name for this quantity. It can probably be thought of similarly to the fractal dimension. –Joke 20:55, 6 October 2006 (UTC)